Negligibility of Projective Linear Automorphisms

Ahmad, Hamza; Hajja, Mowaffaq; Kang, Ming-chang

doi:10.1006/jabr.1997.7199

Cited by 3 publications

(3 citation statements)

References 5 publications

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“…The notion of negligibility was introduced in [6], and several negligibility theorems, together with applications to rationality problems can be found in [1,7,11,12], and [2]. In particular, it follows from [8] and [9] that if f and g are minimal polynomials of the linear k-automorphisms α and β, respectively, then β is negligible relative to α in the two cases when g divides f and when order(g) divides order(f ).…”

Section: Question 1 What Are the Conditions On F (T ) G(t ) ∈ K[t ]mentioning

confidence: 99%

“…, x n ∈ K such that σ (x i ) = a i x i . If c ∈ A, then c is of the form a e 1 1 · · · a e n n , where e i ∈ Z. Letting z = x e 1 1 · · · x e n n , it is easily seen that σ (z) = cz.…”

Section: Lemma 52 Let σ Be the K-automorphism Of K ∈ E(k) Defined Bymentioning

confidence: 99%

“…(12), γ (e) ∈ R γ 2 (1) , and since the action of Γ on B is transitive, R t is non-empty for all t = 1, . .…”

mentioning

confidence: 99%

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Relative negligibility of linear automorphisms

Ahmad

Hajja

2008

Journal of Algebra

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Let U and V be finite-dimensional vector spaces over a field k, α ∈ GL(U ), β ∈ GL(V ) and I be the identity transformation on V . Denote by α * β and α * I the induced linear automorphisms on U ⊕ V ; α * β and α * I can also be regarded as k-automorphisms on the function field k(U ⊕ V ). It is elementary to check whether α * β and α * I are conjugate within GL(U ⊕ V ) by examining their rational canonical forms. In this paper we shall give necessary and sufficient conditions for α * β and α * I to be conjugate within Aut k (k(U ⊕ V )). For this characterization, we introduce the concept of the generalized order. Through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field.

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Section: Question 1 What Are the Conditions On F (T ) G(t ) ∈ K[t ]mentioning

confidence: 99%

Section: Lemma 52 Let σ Be the K-automorphism Of K ∈ E(k) Defined Bymentioning

confidence: 99%

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Relative negligibility of linear automorphisms

Ahmad

Hajja

2008

Journal of Algebra

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Rationality of Some Projective Linear Actions

2000

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On the Simplest Quartic Fields and Related Thue Equations

Hoshi

2014

Computer Mathematics

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Let K be a field of char K = 2. For a ∈ K, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial X 4 −aX 3 −6X 2 +aX +1 over K as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field K, we see that the polynomial gives the same splitting field over K for infinitely many values a of K. We also see by Siegel's theorem for curves of genus zero that only finitely many algebraic integers a ∈ OK in a number field K may give the same splitting field. By applying the result over the field Q of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equationswhere m ∈ Z is a rational integer and c is a divisor of 4(m 2 + 16), and isomorphism classes of the simplest quartic fields.

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Negligibility of Projective Linear Automorphisms

Cited by 3 publications

References 5 publications

Relative negligibility of linear automorphisms

Relative negligibility of linear automorphisms

Rationality of Some Projective Linear Actions

On the Simplest Quartic Fields and Related Thue Equations

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